Subgroups — #11786018

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63646	C₂	1	1	yes	maximal, characteristic
2	C₃×#5106	C₂	1	1	yes	maximal, characteristic
3	C₃×#5101	C₂	1	1	yes	maximal, characteristic
4	#10589169	C₂	1	2	yes	maximal
5	#10589157	C₂	1	2	yes	maximal
6	#70657	—	3	1	no	maximal
7	#69896	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3748	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	2	yes	—
11	C₃×#926	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	#63646	—	2	1	no	—
14	#63646	—	2	1	no	—
15	C₃×#818	—	2	1	no	—
16	C₃×#817	—	2	1	no	—
17	C₃×#815	C₄	1	2	yes	complemented
18	C₃×C₈×Dic₃	C₄	1	2	yes	complemented
19	C₃×#427	—	3	1	no	—
20	C₃×#405	—	3	1	no	—
21	C₂×#819	—	3	1	no	—
22	C₂×#819	—	3	1	no	—
23	C₂×#819	—	6	1	no	—
24	#5106	D₃	1	1	yes	characteristic
25	#5101	D₃	1	1	yes	characteristic
26	#4556	—	3	2	no	—
27	#4533	—	3	2	no	—
28	#4531	—	3	2	no	—
29	#4531	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
37	#3748	—	2	2	no	—
38	#3748	—	2	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃×#217	—	2	1	no	—
41	C₃×#217	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	1	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄×Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
46	C₃×#208	—	2	1	no	—
47	C₃×#208	—	2	1	no	—
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	#1833	—	9	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	1	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×#117	—	3	1	no	—
55	C₃×#117	—	3	2	no	—
56	#926	C₂×D₃	1	2	yes	—
57	#926	C₂×D₃	1	1	yes	characteristic
58	C₂×#208	—	3	2	no	—
59	C₂×#208	—	3	2	no	—
60	C₂×#208	—	6	2	no	—
61	C₂×#208	C₂×D₃	1	1	yes	characteristic
62	C₃×#107	—	6	1	no	—
63	C₃×#106	—	6	1	no	—
64	C₃×#97	—	3	2	no	—
65	C₃×C₄×C₈	—	3	2	no	—
66	#819	—	6	2	no	—
67	#819	—	6	2	no	—
68	#819	—	6	1	no	—
69	#819	—	6	1	no	—
70	#819	—	6	1	no	—
71	#819	—	6	1	no	—
72	#818	—	2	1	no	—
73	#817	—	2	1	no	—
74	#815	Dic₃	1	2	yes	complemented
75	C₈×Dic₃	Dic₃	1	2	yes	complemented
76	C₂³×C₃²	#41	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	1	yes	characteristic
83	C₂×C₃×Dic₃	—	2	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₃⋊(C₃⋊C₈)	—	4	2	no	—
90	C₃×C₃⋊C₈	—	2	2	no	—
91	C₂×#108	—	9	1	no	—
92	#427	—	3	1	no	—
93	#405	—	3	1	no	—
94	#344	—	9	2	no	—
95	H(3,ℤ/4)	—	9	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₃×M₁₆	—	6	1	no	—
100	C₃×M₁₆	—	6	1	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	3	2	no	—
108	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
109	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
110	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
111	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
112	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
113	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
118	C₂×C₃×C₈	#67	1	1	yes	characteristic
119	C₂×C₃×C₈	#67	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	6	1	no	—
121	C₃×C₄⋊C₄	—	6	2	no	—
122	C₃×C₂²⋊C₄	—	6	1	no	—
123	C₃×C₂²⋊C₄	—	6	2	no	—
124	C₃×C₄²	—	3	2	no	—
125	C₃×C₄²	—	6	2	no	—
126	#217	—	2	1	no	—
127	#217	—	2	2	no	—
128	C₄⋊Dic₃	—	2	1	no	—
129	C₄⋊Dic₃	—	2	2	no	—
130	C₄×Dic₃	—	2	2	no	—
131	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
132	#208	—	2	1	no	—
133	#208	—	2	1	no	—
134	#208	—	6	4	no	—
135	#208	—	6	2	no	—
136	#208	—	6	2	no	—
137	#208	—	6	2	no	—
138	#208	—	6	2	no	—
139	#208	—	12	2	no	—
140	C₂×C₃⋊C₈	—	6	2	no	—
141	C₂×C₃⋊C₈	—	6	2	no	—
142	C₂×C₃⋊C₈	—	12	2	no	—
143	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
144	C₂²×C₃²	—	4	1	no	—
145	C₂²×C₃²	#117	1	1	yes	characteristic
146	C₂²×C₃²	#124	1	1	yes	characteristic
147	C₂²×C₃²	#125	1	1	yes	characteristic
148	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₃²×C₄	#117	1	1	yes	characteristic
150	C₃²×C₄	#124	1	1	yes	characteristic
151	C₃²×C₄	#125	1	1	yes	characteristic
152	C₃×Dic₃	—	4	2	no	—
153	C₃×Dic₃	—	8	2	no	—
154	C₂×M₁₆	—	3	1	no	—
155	C₂×M₁₆	—	9	2	no	—
156	C₂²×C₈	D₃²	1	1	yes	characteristic
157	#117	—	3	1	no	—
158	#117	—	3	2	no	—
159	#108	—	18	1	no	—
160	#108	—	18	1	no	—
161	#107	—	6	1	no	—
162	#106	—	6	1	no	—
163	#97	—	3	2	no	—
164	C₄×C₈	—	3	2	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	#235	1	1	yes	characteristic
167	C₂³×C₃	#237	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	—	12	2	no	—
176	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₄	#235	1	1	yes	characteristic
181	C₂×C₃×C₄	#235	1	1	yes	characteristic
182	C₂×C₃×C₄	#235	1	1	yes	characteristic
183	C₂×C₃×C₄	#237	1	1	yes	characteristic
184	C₂×C₃×C₄	#239	1	1	yes	characteristic
185	C₂×C₃×C₄	#239	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
188	C₂×Dic₃	—	2	2	no	—
189	C₂×Dic₃	—	4	2	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	1	no	—
192	C₃×C₈	—	2	1	no	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	6	2	no	—
195	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
196	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
197	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
198	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
199	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
200	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
201	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
202	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
203	C₃⋊C₈	—	2	2	no	—
204	C₃⋊C₈	—	12	4	no	—
205	C₃⋊C₈	—	12	2	no	—
206	C₃⋊C₈	—	12	2	no	—
207	C₂×C₃²	—	2	1	no	—
208	C₂×C₃²	—	2	1	no	—
209	C₂×C₃²	—	2	1	no	—
210	C₂×C₃²	#382	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
212	M₁₆	—	6	1	no	—
213	M₁₆	—	6	1	no	—
214	M₁₆	—	18	2	no	—
215	M₁₆	—	18	2	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	18	2	no	—
218	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
219	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
220	C₂×C₈	#626	1	1	yes	characteristic
221	C₂×C₈	#626	1	1	yes	characteristic
222	C₂×C₈	#627	1	1	yes	characteristic
223	C₂×C₈	#627	1	1	yes	characteristic
224	C₄⋊C₄	—	6	1	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₂²⋊C₄	—	6	1	no	—
227	C₂²⋊C₄	—	6	2	no	—
228	C₄²	—	3	2	no	—
229	C₄²	—	6	2	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	4	1	no	—
234	C₂²×C₃	—	4	1	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	#870	1	1	yes	characteristic
237	C₂²×C₃	#875	1	1	yes	characteristic
238	C₂²×C₃	#890	1	1	yes	characteristic
239	C₂²×C₃	#892	1	1	yes	characteristic
240	C₂²×C₃	#926	1	1	yes	characteristic
241	C₂²×C₃	#936	1	1	yes	characteristic
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	12	2	no	—
247	C₃×C₄	—	24	2	no	—
248	C₃×C₄	#870	1	1	yes	characteristic
249	C₃×C₄	#875	1	1	yes	characteristic
250	C₃×C₄	#890	1	1	yes	characteristic
251	C₃×C₄	#892	1	1	yes	characteristic
252	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
253	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
254	C₃×C₄	#926	1	1	yes	characteristic
255	C₃×C₄	#947	1	1	yes	characteristic
256	Dic₃	—	4	2	no	—
257	Dic₃	—	8	2	no	—
258	C₃²	#1833	1	1	yes	characteristic, complemented, Sylow
259	C₂³	#3795	1	1	yes	characteristic
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	—	12	2	no	—
262	C₂×C₄	#3795	1	1	yes	characteristic
263	C₂×C₄	#3799	1	1	yes	characteristic
264	C₂×C₄	#3799	1	1	yes	characteristic
265	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#626	1	1	yes	characteristic
267	C₂×C₄	C₂×#627	1	1	yes	characteristic
268	C₈	—	6	2	no	—
269	C₈	—	36	2	no	—
270	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
271	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
272	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
273	C₈	Dic₃⋊Dic₃	1	1	yes	characteristic
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	4	1	no	—
283	C₂×C₃	—	4	1	no	—
284	C₂×C₃	#4936	1	1	yes	characteristic
285	C₂×C₃	#4977	1	1	yes	characteristic
286	C₂²	—	4	1	no	—
287	C₂²	#63841	1	1	yes	characteristic
288	C₂²	#63860	1	1	yes	characteristic
289	C₂²	#63869	1	1	yes	characteristic
290	C₄	—	12	2	no	—
291	C₄	—	24	2	no	—
292	C₄	#63841	1	1	yes	characteristic
293	C₄	#63864	1	1	yes	characteristic
294	C₄	#63869	1	1	yes	characteristic
295	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
296	C₃	—	2	1	no	—
297	C₃	#69896	1	1	yes	characteristic, complemented
298	C₃	#70657	1	1	yes	characteristic, complemented
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#10591224	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63841
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂×C₃²	#382
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1833
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
17. C₃×#815	C₄	semidirect
18. C₃×C₈×Dic₃	C₄	semidirect
74. #815	Dic₃	semidirect
75. C₈×Dic₃	Dic₃	semidirect
258. C₃²	#1833	semidirect
297. C₃	#69896	semidirect
298. C₃	#70657	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#63646	2
C₃×#5106	2
C₃×#5101	2
#10589169	2
#10589157	2
#70657	3
#69896	3